Within the scenario just discussed, we model the merger rate as [1]}:
\(\frac{\mathrm {d}^2\tau _{\rm merg}(z,\,m_1,\,m_2)}{\mathrm {d}\log _{10} m_1\,\mathrm {d}\log _{10} m_2}=\overbrace{R_{\rm clust}\frac{(m_1 + m_2)^{10/7} }{(m_1 m_2)^{5/7}}\,\frac{(1+z)^{\alpha _z}}{\left[1+\left(M_{\rm tot}/M_*\right)\right]^{\alpha _c}}}^{\rm PBH\,clustering/binary\, formation}\,\overbrace{f_{\rm PBH}(m_1)f_{\rm PBH}(m_2)}^{\rm primordial / thermal\, evolution}.\) 
